Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(fact(X)) → IF(zero(X), s(0), prod(X, fact(p(X))))
ACTIVE(fact(X)) → ZERO(X)
ACTIVE(fact(X)) → S(0)
ACTIVE(fact(X)) → PROD(X, fact(p(X)))
ACTIVE(fact(X)) → FACT(p(X))
ACTIVE(fact(X)) → P(X)
ACTIVE(add(s(X), Y)) → S(add(X, Y))
ACTIVE(add(s(X), Y)) → ADD(X, Y)
ACTIVE(prod(s(X), Y)) → ADD(Y, prod(X, Y))
ACTIVE(prod(s(X), Y)) → PROD(X, Y)
ACTIVE(fact(X)) → FACT(active(X))
ACTIVE(fact(X)) → ACTIVE(X)
ACTIVE(if(X1, X2, X3)) → IF(active(X1), X2, X3)
ACTIVE(if(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(zero(X)) → ZERO(active(X))
ACTIVE(zero(X)) → ACTIVE(X)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(prod(X1, X2)) → PROD(active(X1), X2)
ACTIVE(prod(X1, X2)) → ACTIVE(X1)
ACTIVE(prod(X1, X2)) → PROD(X1, active(X2))
ACTIVE(prod(X1, X2)) → ACTIVE(X2)
ACTIVE(p(X)) → P(active(X))
ACTIVE(p(X)) → ACTIVE(X)
ACTIVE(add(X1, X2)) → ADD(active(X1), X2)
ACTIVE(add(X1, X2)) → ACTIVE(X1)
ACTIVE(add(X1, X2)) → ADD(X1, active(X2))
ACTIVE(add(X1, X2)) → ACTIVE(X2)
FACT(mark(X)) → FACT(X)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)
ZERO(mark(X)) → ZERO(X)
S(mark(X)) → S(X)
PROD(mark(X1), X2) → PROD(X1, X2)
PROD(X1, mark(X2)) → PROD(X1, X2)
P(mark(X)) → P(X)
ADD(mark(X1), X2) → ADD(X1, X2)
ADD(X1, mark(X2)) → ADD(X1, X2)
PROPER(fact(X)) → FACT(proper(X))
PROPER(fact(X)) → PROPER(X)
PROPER(if(X1, X2, X3)) → IF(proper(X1), proper(X2), proper(X3))
PROPER(if(X1, X2, X3)) → PROPER(X1)
PROPER(if(X1, X2, X3)) → PROPER(X2)
PROPER(if(X1, X2, X3)) → PROPER(X3)
PROPER(zero(X)) → ZERO(proper(X))
PROPER(zero(X)) → PROPER(X)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(prod(X1, X2)) → PROD(proper(X1), proper(X2))
PROPER(prod(X1, X2)) → PROPER(X1)
PROPER(prod(X1, X2)) → PROPER(X2)
PROPER(p(X)) → P(proper(X))
PROPER(p(X)) → PROPER(X)
PROPER(add(X1, X2)) → ADD(proper(X1), proper(X2))
PROPER(add(X1, X2)) → PROPER(X1)
PROPER(add(X1, X2)) → PROPER(X2)
FACT(ok(X)) → FACT(X)
IF(ok(X1), ok(X2), ok(X3)) → IF(X1, X2, X3)
ZERO(ok(X)) → ZERO(X)
S(ok(X)) → S(X)
PROD(ok(X1), ok(X2)) → PROD(X1, X2)
P(ok(X)) → P(X)
ADD(ok(X1), ok(X2)) → ADD(X1, X2)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 10 SCCs with 28 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ADD(X1, mark(X2)) → ADD(X1, X2)
ADD(mark(X1), X2) → ADD(X1, X2)
ADD(ok(X1), ok(X2)) → ADD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ADD(mark(X1), X2) → ADD(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ADD(x1, x2)  =  ADD(x1)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > ADD₁

Status:

mark₁: [1]
ADD₁: [1]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ADD(X1, mark(X2)) → ADD(X1, X2)
ADD(ok(X1), ok(X2)) → ADD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ADD(ok(X1), ok(X2)) → ADD(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ADD(x1, x2)  =  ADD(x1)
mark(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]
ADD₁: [1]

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ADD(X1, mark(X2)) → ADD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ADD(X1, mark(X2)) → ADD(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > ADD₂

Status:

mark₁: [1]
ADD₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(13) TRUE

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P(ok(X)) → P(X)
P(mark(X)) → P(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P(ok(X)) → P(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
P(x1)  =  P(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > P₁

Status:

P₁: [1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P(mark(X)) → P(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P(mark(X)) → P(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
P(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(20) TRUE

(21) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROD(X1, mark(X2)) → PROD(X1, X2)
PROD(mark(X1), X2) → PROD(X1, X2)
PROD(ok(X1), ok(X2)) → PROD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROD(mark(X1), X2) → PROD(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROD(x1, x2)  =  PROD(x1)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > PROD₁

Status:

PROD₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROD(X1, mark(X2)) → PROD(X1, X2)
PROD(ok(X1), ok(X2)) → PROD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROD(ok(X1), ok(X2)) → PROD(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROD(x1, x2)  =  PROD(x1)
mark(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

PROD₁: [1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROD(X1, mark(X2)) → PROD(X1, X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROD(X1, mark(X2)) → PROD(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > PROD₂

Status:

mark₁: [1]
PROD₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(28) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(29) TRUE

(30) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(ok(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > S₁

Status:

ok₁: [1]
S₁: [1]

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(35) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(36) TRUE

(37) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ZERO(ok(X)) → ZERO(X)
ZERO(mark(X)) → ZERO(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ZERO(ok(X)) → ZERO(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZERO(x1)  =  ZERO(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > ZERO₁

Status:

ZERO₁: [1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ZERO(mark(X)) → ZERO(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ZERO(mark(X)) → ZERO(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZERO(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(42) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(43) TRUE

(44) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IF(ok(X1), ok(X2), ok(X3)) → IF(X1, X2, X3)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

IF(ok(X1), ok(X2), ok(X3)) → IF(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3)  =  IF(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > IF₁

Status:

ok₁: [1]
IF₁: [1]

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IF(mark(X1), X2, X3) → IF(X1, X2, X3)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

IF(mark(X1), X2, X3) → IF(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(49) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(50) TRUE

(51) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FACT(ok(X)) → FACT(X)
FACT(mark(X)) → FACT(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FACT(ok(X)) → FACT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FACT(x1)  =  FACT(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > FACT₁

Status:

FACT₁: [1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FACT(mark(X)) → FACT(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FACT(mark(X)) → FACT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FACT(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(56) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(57) TRUE

(58) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(if(X1, X2, X3)) → PROPER(X1)
PROPER(fact(X)) → PROPER(X)
PROPER(if(X1, X2, X3)) → PROPER(X2)
PROPER(if(X1, X2, X3)) → PROPER(X3)
PROPER(zero(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(prod(X1, X2)) → PROPER(X1)
PROPER(prod(X1, X2)) → PROPER(X2)
PROPER(p(X)) → PROPER(X)
PROPER(add(X1, X2)) → PROPER(X1)
PROPER(add(X1, X2)) → PROPER(X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(if(X1, X2, X3)) → PROPER(X1)
PROPER(if(X1, X2, X3)) → PROPER(X2)
PROPER(if(X1, X2, X3)) → PROPER(X3)
PROPER(prod(X1, X2)) → PROPER(X1)
PROPER(prod(X1, X2)) → PROPER(X2)
PROPER(add(X1, X2)) → PROPER(X1)
PROPER(add(X1, X2)) → PROPER(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
if(x1, x2, x3)  =  if(x1, x2, x3)
fact(x1)  =  x1
zero(x1)  =  x1
s(x1)  =  x1
prod(x1, x2)  =  prod(x1, x2)
p(x1)  =  x1
add(x1, x2)  =  add(x1, x2)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

prod₂: [1,2]
add₂: [1,2]
PROPER₁: [1]
if₃: [1,2,3]

The following usable rules [FROCOS05] were oriented: none

(60) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(fact(X)) → PROPER(X)
PROPER(zero(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(p(X)) → PROPER(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(fact(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
fact(x1)  =  fact(x1)
zero(x1)  =  x1
s(x1)  =  x1
p(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

fact₁ > PROPER₁

Status:

PROPER₁: [1]
fact₁: [1]

The following usable rules [FROCOS05] were oriented: none

(62) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(zero(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(p(X)) → PROPER(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(p(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
zero(x1)  =  x1
s(x1)  =  x1
p(x1)  =  p(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

PROPER₁: [1]
p₁: [1]

The following usable rules [FROCOS05] were oriented: none

(64) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(zero(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(zero(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
zero(x1)  =  zero(x1)
s(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

zero₁ > PROPER₁

Status:

zero₁: [1]
PROPER₁: [1]

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(s(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
s(x1)  =  s(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

s₁: [1]

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(69) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(70) TRUE

(71) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(if(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(fact(X)) → ACTIVE(X)
ACTIVE(zero(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(prod(X1, X2)) → ACTIVE(X1)
ACTIVE(prod(X1, X2)) → ACTIVE(X2)
ACTIVE(p(X)) → ACTIVE(X)
ACTIVE(add(X1, X2)) → ACTIVE(X1)
ACTIVE(add(X1, X2)) → ACTIVE(X2)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(prod(X1, X2)) → ACTIVE(X1)
ACTIVE(prod(X1, X2)) → ACTIVE(X2)
ACTIVE(add(X1, X2)) → ACTIVE(X1)
ACTIVE(add(X1, X2)) → ACTIVE(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
if(x1, x2, x3)  =  x1
fact(x1)  =  x1
zero(x1)  =  x1
s(x1)  =  x1
prod(x1, x2)  =  prod(x1, x2)
p(x1)  =  x1
add(x1, x2)  =  add(x1, x2)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

prod₂: [1,2]
add₂: [1,2]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(73) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(if(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(fact(X)) → ACTIVE(X)
ACTIVE(zero(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(p(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(if(X1, X2, X3)) → ACTIVE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
if(x1, x2, x3)  =  if(x1, x2, x3)
fact(x1)  =  x1
zero(x1)  =  x1
s(x1)  =  x1
p(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

if₃ > ACTIVE₁

Status:

if₃: [1,2,3]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(fact(X)) → ACTIVE(X)
ACTIVE(zero(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(p(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(fact(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
fact(x1)  =  fact(x1)
zero(x1)  =  x1
s(x1)  =  x1
p(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

fact₁ > ACTIVE₁

Status:

fact₁: [1]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(77) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(zero(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(p(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(78) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(p(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
zero(x1)  =  x1
s(x1)  =  x1
p(x1)  =  p(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

p₁: [1]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(79) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(zero(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(zero(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
zero(x1)  =  zero(x1)
s(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

zero₁ > ACTIVE₁

Status:

zero₁: [1]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(81) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(s(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(s(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
s(x1)  =  s(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

s₁: [1]

The following usable rules [FROCOS05] were oriented: none

(83) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(84) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(85) TRUE

(86) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.